We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of Entropy Solution of a Burgers' type scalar conservation law on the other. The solution of the former is obtained by $x$-differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient-flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws.

We present a general framework for the numerical resolution of processes
driven by geometric components such as the curvature of the domain boundary.
In this context, the Laplace-Beltrami operator plays a crucial role.
Two applications are discussed: Electrowetting on dielectric (EWOD) and
the simulation of Biomembranes.

The former refers to a parallel-plate
micro-device that moves fluid droplets through electrically
actuated surface tension effects. These devices have potential
applications in biomedical `lab-on-a-chip' devices (such as automated DNA
testing and cell separation) and controlled micro-fluidic transport
(e.g. mixing and concentration control).
We model the fluid
dynamics using Hele-Shaw type equations (in 2-D) with a focus on
including the relevant boundary phenomena. Specifically, we model contact
line pinning as a static (Coulombic) friction effect that effectively
becomes a variational inequality for the motion of
the liquid-gas interface.
We analyze this approach, present simulations and compare them to
experimental videos of EWOD driven droplets.

The latter applies to 2 mono-molecular forming an encapsulating bag called
vesicle. Equilibrium shapes are obtained via the minimization of the
Willmore energy under area and volume constraints. Physical dynamics are
obtained by taking into account the effect of the inside (bulk) fluid.
Forth order, highly nonlinear arising problems are solved using an
adaptive mixed finite element method.
Two and three dimensional simulations are presented.
In particular, typical biconcave shape specific to red blood cells are
obtained.

This presentation is based on joint works with R. Nochetto, M. Pauletti, and S. Walker.

The Kaye effect is a fascinating phenomenon of a leaping shampoo stream
which was first described by Alan Kaye in 1963 as a property of non-Newtonian fluid.
It manifest itself when a thin stream of non-Newtonian fluid is poured into a dish of fluid.
As pouring proceeds, a small stream of liquid occasionally leaps upward from the heap.

Shear-thinning viscosity is advanced as the critical ingredient to understand this effect.
We consider a Carreau-Yasuda model and numerically identify the parameters eventually yielding to the Kaye effect.
The numerical algorithm consists of a projection method coupled with a level-set formulation for the interface representation.
In this talk, we focus on two aspects: (i) the numerical approximation of the capillarity force and (ii) the entropy residual technique used to stabilize the finite element approximation of the level-set evolution.

Fluid-solid interaction (FSI) problems arise in many applications. They include multi-physics problems in engineering such as aeroelasticity and propeller turbines, as well as biofluidic application such as self-propulsion organisms, fluid-cell interactions, and the interaction between blood flow and cardiovascular tissue. A comprehensive study of these problems remains to be a challenge due to their strong nonlinearity and multi-physics nature. To make things worse, in many biological applications the solid is composed of several layers, each with different mechanical characteristics. This is, for example, the case with arterial walls whose physiology and pathophysiology are affected by their interaction with blood flow, and by the interaction between the different arterial wall layers. In this introductory lecture, the speaker will present an overview of the main problems in FSI in blood flow motivated by cardiovascular applications. Models for blood flow and arterial walls will be presented. The coupling between an incompressible, viscous fluid modeling blood flow, and an elastic/viscoelastic solid modeling arterial walls will be described. The geometric nonlinearities in the coupled problem will be discussed, and the reasons for the instabilities in the classical partitioned (loosely-coupled) FSI schemes will be explained. A new recent result showing that an effective boundary condition at the fluid-elastic solid interface in the blood flow application, is the slip-condition, will be discussed.

The development of existence theory for moving boundary, fluid-solid interaction problems involving a viscous, incompressible fluid has become particularly active since the 1990's. The first existence results were obtained for the cases in which the structure is completely immersed in the fluid, and the structure was considered to be either a rigid body, or described by a finite number of modal functions. The results that followed concerned the coupling between the Navier-Stokes equations and $2D$ or $3D$ linear elasticity, where the coupling was assumed on a fixed fluid-structure interface (linear coupling). The first existence result that considered the coupling between an incompressible, viscous fluid and an elastic structure, where the coupling was assumed to take place at a deformed fluid-structure interface (nonlinear coupling), was obtained in 2004 by B. daVeiga. Since then, several works in this area, considering different structural models, and different mathematical techniques, were obtained. Recently, the speaker, together with collaborator Muha, obtained the first constructive existence proof for a benchmark FSI problem in blood flow, which was recently extended to study fluid-multi-layered structure interaction problems. The main steps in the proof rely on a loosely-coupled numerical scheme, in which the fluid and the structure sub-problems are separated by an operator-splitting approach. In this lecture, the speaker will give an overview of the existing literature on the existence analysis for FSI problems, and will show the main ideas behind the constructive proof of the existence of a solution to a benchmark problem in FSI in blood flow. A new result that indicates that the presence of a thin fluid-structure interface with mass regularizes solutions to this FSI problem, will be discussed.

The development of numerical solvers for fluid-elastic solid interaction problems has become particularly active since the 1980's. Among the most popular techniques are the Immersed Boundary method and the Arbitrary Lagrangian Eulerian method. We further mention the Fictitious Domain method, the Lattice Boltzmann method, the Level Set method, and the Coupled Momentum method.
Until recently, only monolithic algorithms seemed applicable to blood flow simulations. These algorithms are based on solving the entire nonlinear, coupled problem as one monolithic system. They are, however, generally quite expensive in terms of computational time, programming time and memory requirements, since they require solving a sequence of strongly coupled problems using, e.g., the fixed point and Newton's methods. The multi-physics nature of the blood flow problem strongly suggests employing partitioned (or staggered) numerical algorithms, where the coupled fluid-structure interaction problem is separated into a fluid and a structure sub-problem. The fluid and structure sub-problems are integrated in time in an alternating way, and the coupling conditions are enforced asynchronously. These classical (Dirichlet-Neumann loosely-coupled) partitioned schemes work well for problems in aeroelasticity. Unfortunately, when the fluid and structure have comparable densities, which is the case in the blood flow application, the simple strategy of separating the fluid from the structure suffers from severe stability issues. To get around these difficulties, and to retain the main advantages of loosely-coupled partitioned schemes such as modularity, simple implementation, and low computational costs, several new loosely-coupled algorithms have been proposed recently. In this lecture we will present a short overview of the basic numerical strategies for solving fluid-elastic solid interaction in blood flow, and will present the core ideas behind a novel, stable, loosely-coupled scheme, recently introduced by the speaker and collaborators to numerically simulate a class of multi-physics problems arising in blood flow applications. A recent extension of the scheme to study fluid-multi-layered structure interaction problems in blood flow will also be presented.

Motility of cells is at the root of many fundamental processes in biology: from sperm cells swimming to fertilize an egg cell, to metastatic tumor cells crawling to invade nearby tissues.

We will discuss the mechanical bases of cellular motility by swimming and crawling. Special emphasis will be placed on the connections between low Reynolds number swimming and Geometric Control Theory, and on the geometric structure of the underlying equations of motion.

As a concrete example, we will report on reverse engineering of the euglenoid movement. The lessons learned in the context of swimming motility will be then applied to selected case studies of crawling motility.

Our goal in this talk is to derive a method to compute surfaces that minimize general surface energies, in the form of weighted surface integrals with weights depending on the normal and the curvature of the surface. Energies of this form have applications in many areas, such as material science, biology and image processing. A well-known example of such energies is the Willmore functional, which is an integral of the squared mean curvature and is used as a regularization term in surface restoration and a model for the bending energy of vesicle membranes. Other examples are the weighted Willmore functional, the energy with spontaneous curvature and anisotropic surface energies. In this talk, I will derive the first variation of the general curvature-dependent surface energy using shape differential calculus. The first variation clearly delineates the influence of various components on the energy and can be used to write a gradient descent flow to compute surfaces that minimize the surface energy. However, it is difficult to discretize and implement numerically. Therefore, I will derive an alternate weak form. The weak form relies on computable quantities of the surface, and in particular, avoids tangential differentiation of the unit normal. I will use this new formulation to devise a finite element discretization of the gradient descent algorithm.

Many natural phenomena and engineering problems can be modeled as shape optimization problems, in which our goal is to find shapes, such as curves in 2d or surfaces in 3d, minimizing certain shape energies. Examples of such problems are modeling of crystalline interfaces in material science, vesicles in biology, and image segmentation in computer vision. Finding computational solutions for such problems requires performing an optimization over a space of candidate shapes. In this talk, I will introduce a methodology to compute the optimal shapes in such scenarios. I will first describe how we can use shape calculus to derive a gradient descent flow for a given shape energy. The gradient descent flow can be used to deform a given initial shape towards a minimum of the energy in an iterative manner. I will describe how to discretize the gradient descent flow using the finite element method. Then I will explain how these ingredients can be put together in an effective shape optimization algorithm. I will underscore the role of errors and adaptivity, and I will explain the role of stopping criterion and step size selection in devising an efficient and robust algorithm. I will demonstrate the effectiveness of the method with several examples from image processing.

Brain tissue is an inhomogeneous, layered, multi-phase, and multi-functional material made of interconnected neural, glial, and vascular networks, and is immersed in the cerebrospinal fluid. In order to advance our understanding of how the brain provides its functions, we need to develop a robust controlled feedback engineering framework that uses fundamental science concepts to guide and interpret experiments investigating brain’s response to different types of stimuli, aging, trauma, diseases, treatment and recovery processes. Improved multi-physics constitutive models are needed that account for the complex heterogeneity and dynamics of the material brain, differentiate between healthy and diseased tissues, as well as include the mechano-chemistry regulating brain’s functions. In this context, I will present some mathematical models of brain neuro-mechanics and corresponding numerical solvers that I and my collaborators have been developing over the last decade. I will also discuss the use of fractional calculus in modeling multiple temporal scales and non-locality of brain dynamics. Lastly, I will present our generalization of the continuum mechanics theory using fractional calculus.

We begin the tutorial by introducing a couple of interface
problems in various physical and biological applications as
motivating examples. We then present some basic ingredients
of phase field models and take a special example to illustrate
both analytical and computational aspects of the relevant
variational problems and associated gradient dynamics.
We discuss in particular dynamics related to the classical
Allen-Cahn model and Cahn-Hilliard models with various
mobilities. We next focus on a few phase field models for
selected applications in biology and materials science which
are developed for interface problems involving elastic energy
contributions and those involving the coupling with external fields.
Finally, we conclude the tutorial with discussions on other
issues related to phase field modeling and simulations such as
robust retrieval of statistical information, stochastic fluctuation
and effective multiscale modeling.

The lectures will begin with an overview of the various methods used in the simulations of model biomembranes and lipid bilayer vesicles at different scales. Then, the course will focus on discussing models and simulations at the continuum scales. In particular, the lecturer will discuss the bending elasticity models for the membrane deformation and their formulations in terms of the phase field calculus/diffuse interface approach. Techniques will be introduced on how to incorporate various interesting features such as adhesion, multicomponent phase separation, vesicle fusion, hydrodynamic interactions and thermal fluctuations. In addition, numerical methods will be presented that can
be used to improve the efficiency, reliability and predictability of the numerical simulations.

In the lectures, I will first introduce and review several chemotaxis models including the classical Patlak-Keller-Segel model. Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. Chemotaxis is an important process in many medical and biological applications including bacteria/cell aggregation, pattern formation mechanisms, and tumor growth.

The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of partial differential equations (PDEs). Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. This blow-up represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems. In either case, capturing such solutions numerically is a challenging problem.

In the lectures, we will conduct a detailed review of several recently developed numerical methods for the approximation and simulation of the chemotaxis and related models. In particular, we will discuss several finite element, finite-volume and hybrid finite-volume-finite-difference methods, which are designed to achieve an ultimate goal of developing a highly accurate, stable and robust numerical methods for chemotaxis models.

Paolo Biscari and Stefano Turzi considered a plate with an undulatory pattern. They replace the corrugation with sinusoidal boundary conditions, and use formal asymptotics for the analysis. I would like to use the method of gamma convergence to determine the effective energy and its minimizers for this problem.

Coatings made from colloidal suspensions play a central role
in a broad range of applications such as zeolite-based separation membranes, templates for photonic materials,
printed electronics, and biomedical engineering. Uniform coatings are often desired, but these can be difficult to achieve
due to various phenomena such as the well-known coffee-ring effect. We are interested in developing a fundamental understanding
of the factors that determine whether a coating is uniform, and in this talk I will discuss
our work on two problems concerning this issue. The first problem involves development of a mathematical model for the drying
of a droplet laden with colloidal particles. In contrast to prior work, our model accounts for depthwise gradients in particle
concentration, and can thus qualitatively capture the formation of colloidal "skins" that are observed experimentally. The
second problem involves an experimental study of the dip coating of nanoparticle suspensions. We demonstrate that there is a
"sweet spot" in the parameter space of solids concentration and withdrawal speed where monolayer coatings can be achieved.
Analysis of the experimental data suggests that in order to obtain such coatings, the substrate needs to be withdrawn rapidly
enough to overcome pinning of the liquid-air meniscus on the particles, but not so rapidly that a continuous liquid film is entrained.

High-speed printing processes are a leading technology
for the large-scale manufacture of a new generation of
nanoscale and microscale devices. Central to all printing
processes is the transfer of liquid from one surface to
another, a seemingly simple unit operation that is still not
well-understood. We will discuss how theory and experiment
can be successfully employed to shed light on the mechanics
of liquid transfer. The insights gained suggest
strategies for engineering the interfacial behavior that
lies at the heart of the liquid transfer operation.

This course will be concerned with the analysis of thin elastic films which exhibit residual stress at free equilibria. Examples of such structures include, in particular, growing tissues such as leaves, flowers or marine invertebrates, as well as specifically engineered gels. There, it is conjectured that the growth process results in the formation of non-Euclidean target metrics, leading to complicated morphogenesis of the tissue which attains an orientation-preserving configuration closest possible to an isometric immersion of the metric.
This phenomenon can be studied through a variational model, pertaining to the non-Euclidean version of the nonlinear elasticity. For metrics with non-zero Riemann curvature, the infimum of the energy turns out to be positive at free equilibria. Further analysis of scaling of the energy minimizers in terms of the vanishing reference plate's thickness leads to the rigorous derivation of the corresponding limiting theories. These theories are differentiated by the embeddability properties of the target metrics - in the same spirit as different scalings of external forces lead to a hierarchy of nonlinear plate theories in classical elasticity whose rigorous (ansatz-free) derivation has been given by Friesecke, James and Muller.
The course will be self-contained and at a level suitable for PhD students having some familiarity with Mathematical Analysis. No prior experience with the research topics under discussion will be expected.

We present a new model concept for pre-diction of boundary layer transition using a linear eddy-viscosity RANS approach. It is a single-point, phys-ics-based method that adopts an alternative to the Laminar Kinetic Energy (LKE) framework. The model is based on a description of the transition process previously discussed by Walters (2009). The version of the model presented here uses the k-omega SST model as the baseline, and includes the effects of transition through one additional transport equation for v2. Here v2 is interpreted as the energy of fully turbulent, 3D velocity fluctuations, while k represents the energy of both fully turbulent and pre-transitional velocity fluctuations. This modeling approach leads to slow growth of fluctuating energy in the pre-transitional region and relaxation towards a fully turbulent model result downstream of transition. Simplicity of the formulation and ease of extension to other baseline models are two potential advantages of the new method. An initial version of the model has been implemented as a UDF subroutine in the commercial CFD code FLUENT and tested for canonical flat plate boundary layer test cases with different freestream turbulence conditions.

Electrolyte and cell volume regulation is essential in physiological systems.
After a brief introduction to cell volume control and electrophysiology,
I will discuss the classical pump-leak model of electrolyte and cell volume control.
I will then generalize this to a PDE model that allows for the
modeling of tissue-level electrodiffusive and osmotic phenomena.
This model will then be applied to the study of cortical spreading depression,
a pathophysiological phenomenon in the brain that is thought to underlie
migraine aura and other pathologies.

We begin by analyzing some simple models for Chlamydomonas
reinhardtii, a microscopic organism that swims in the Stokes regime.
Chlamydomonas beats its flagellum at high frequency, creating
oscillations as it moves. One of our goals is to determine how
important these oscillations are to the effect diffusivity caused by
swimming. Later on we will explore larger organisms, like copepods,
that escape the Stokes regime and additionally have more complex fluid
dynamics.

We present a comprehensive approach to the formulation and discretization of geometric PDE governing processes relevant in biophysics and materials science. We start with key elements of differential geometry and shape differential calculus which enable us to compute first variations of domain and boundary functionals. We propose geometric gradient flows as a relaxation towards equilibrium and derive the corresponding dynamic equations and their finite element approximation. We apply this framework to mean curvature flow, surface diffusion, and Helfrich flow (Willmore flow with area and volume constraints). We compare the dynamics of biomembranes dictated by either relaxation or an incompressible fluid. We examine the coupling of director fields with flexible surfaces, as occur in models of surfactants and gels. We conclude with large bending deformations with isometry constraint and their application to bilayer actuators.

Cell and tissue movement are essential processes at various stages in the life
cycle of multicellular organisms. Early development involves individual and
collective cell movement, leukocytes must migrate toward sites of infection as
part of the immune response, and in cancer directed movement is involved in
invasion and metastasis. The forces needed to drive movement arise from actin
polymerization, molecular motors and other processes, but understanding the
cell- or tissue-level organization of these processes that is needed to produce
the forces necessary for directed movement at the appropriate point in the cell
or tissue is a major challenge. We will focus both on the basic principles and
techniques in continuum mechanics at various levels of organization, as well as
on specific applications such as cell shape and motility, growth and deformation
of tissues, and cell- and tissue-level models of
cancer.

In this course we develop the methodology of “variational modeling” of energy-driven systems. This methodology applies to systems whose evolution (in time) is driven by the decrease of an energy, in a friction-dominated or strongly damped way. Recent developments have shown that a surprisingly large class of evolutionary systems is of this form, even though the energy and the friction mechanism may not be obvious. Examples of these include linear and nonlinear diffusion equations, nonlocal diffusion equations, higher-order parabolic equations, moving-boundary problems, and many others. The course will offer discussions on how to use the mathematical structure as a modeling tool: each choice of an energy and a friction mechanism provide an evolutionary system. The two choices characterize in a remarkably clear way the modeling choices that underlie the resulting differential equations.

We introduce different diffuse approximations of the Willmore energy and the corresponding gradient flows. In the case of well separated interfaces all these flows (formally) converge to the Willmore flow. However, if interfaces collide the different approximations show very different behavior. This will be discussed in view of the Gamma convergence of the corresponding diffuse energies and the existence of so-called saddle solutions of the Allen-Cahn equation. This presentation is based on joint work with S. Esedoglu (Michigan) and A. Rätz (Dortmund).

The objective of this work is the development of a fracture model for brittle polycrystalline materials. The model is based on the discrete element method, and the digital representation of the microstructure composed of a sheet of hexagonal, columnar Voroni polyhedra that represents slab of equal sized close packed grains. Here, we present the initial development of a coarse-grained particle dynamics method to model crack propagation in brittle ceramics at the mesoscale where the effects of grain size and orientation strongly influence the fracture mechanics.

We consider coupled reaction-diffusion models, where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. Classical potential theory and estimates for linear initial boundary value problems are used to prove local well-posedness and global existence. This type of system arises in mathematical models for cell processes.

Hydrogels are crosslinked polymer networks that absorb substantial amounts of water. Cytoplasm and extracelleular matrix and connective tissues can be regarded as natural hydrogels. Synthetic hydrogels based on organic polymers have numerous applications, partly because they can mimic the mechanical, mass transport, and interfacial properties of biological tissues. Following a description of some of these applications, a brief survey will be provided of the ways hydrogels are synthesized and characterized, identifying important structural characteristics. A theory of free swelling of uncharged hydrogels due initially to Flory and Rehner will be reviewed. It will also be shown how the theory of Donnan equilibrium can be adapted to augment Flory-Rehner theory and model the free swelling of polyelectrolyte gels. Interesting and unexpected pattern formations, which occur when swelling is constrained, will also be discussed.

Hydrogels can play both passive and active roles in chemical oscillations and pattern formations. Passively, they can provide a nonconvective environment in which reactions occur, or they can respond by swelling or shrinking according changes in the chemical environment, which is time and/or space varying. In recent years, more active feedback couplings between hydrogels and chemical reactions have been investigated. Hydrogel systems that are coupled to the oscillating Belousov-Zhabotinsky reaction have been well studied. In a second system, a hydrogel separates two volumes, one containing the substrate for an enzyme catalzed reaction, and the other containing the enzyme. The swelling of the hydrogel, and hence transport of the substrate, are controlled by product concentration. With sufficient nonlinear delayed feedback between membrane swelling and enzyme reaction, the system is driven into an oscillatory mode, which can be harnessed to achieve rhythmic hormone release. Experimental evidence and a sequence of increasingly complex mathematical models pertaining to this second system will be presented.

Moving boundary problems lie at heart of most nonequilibrium phase change processes in materials, but also in morphogenesis in biology. Their interest, as well as difficulty, lies in the intricate spatio-temporal patterns that are charateristic of their evolution. We consider in this set of lectures the fundamentals of a mesoscopic description of boundary moving problems, the principles required in the derivation of their transport models, and make a few remarks on computational methodologies to address them.

We present a diffuse interface model for the phenomenon of electrowetting on dielectric and present an analysis of the arising system of equations. Moreover, we study discretization techniques for the problem. The model takes into account different material parameters in each phase and incorporates the most important physical processes, such as incompressibility,electrostatics and dynamic contact lines; all necessary to model the real phenomena. The arising nonlinear system couples the variable density incompressible Navier-Stokes equations for velocity and pressure with a Cahn-Hilliard type equation for the phase variable and chemical potential, a convection diffusion equation for the electric charges and a Poisson equation for the electric potential. Numerical experiments are presented, which show splitting/coalescence of droplets and contact line pinning. This presentation is based on joint work with A. Salgado (U. of Tennessee) and R. Nochetto (U. of Maryland).

Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the well-posedness of the time semi-discrete and fully discrete (finite element) model and discuss error estimates. Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as future work on optimal control of these types of flows.

In this work, we analyze a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentra- tions is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturba- tion theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical po- tentials or voltages for ion size effects. Under electroneutrality (zero net charge) boundary conditions, each of these two critical potentials separates the poten- tial into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary concentrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneu- trality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations and critical potentials agree with those with a nonlocal hard-sphere potential examined by Ji and Liu [J. Dynam. Differential Equations 24 (2012), 955-983].